Question

Show that the sequence $\log a,\log \left(ab\right),\log \left({ab}^3\right),......$ is an A.P. also determine its $nth$ terms.

Answer 1

Considering the series to be:

$\log a,\log \left(ab\right),\log \left(a{b}^2\right),\log \left(a{b}^3\right),.......$

$a=\log a$

$d=\log \left(ab\right)-\log a=\log a+\log b-\log a=\log b$

and $d=\log \left(a{b}^2\right)-\log \left(ab\right)=\log a+\log b^2-(\log a+\log b)=\log a+2\log b-\log a-\log b=\log b$

$\therefore d=\log b$

Therefore the given series is in A.P.

$T_n=a+(n-1)d$

$=\log a+(n-1)\log b$

$=\log a+n\log b-\log b$

$=\log a+\log b^n-\log b$

$=\log a+\log b^{n-1}$

$\therefore T_n=\log a{b}^{n-1}$